Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing variable delay

The purpose of this paper is to obtain the error bounds of fully geometric mesh one-leg methods for solving the nonlinear neutral functional differential equation with a vanishing delay. For this purpose, we consider G(q)-algebraically stable one-leg methods which include the midpoint rule as a special case. The error of the first-step integration implemented by the midpoint rule on [0,T-0] is first estimated. The optimal convergence orders of the fully geometric mesh one-leg methods with respect to T-0 and the mesh diameter hmax are then analyzed and provided for such equation. Numerical studies reported for several test cases confirm our theoretical results and illustrate the effectiveness of the proposed method.